Sur un probl\`eme de compatibilit\'e local-global localement analytique

TL;DR

This paper refines a conjecture by Breuil on locally analytic Ext^1 groups using $(, )$-modules over the Robba ring and proves several special cases, including for ${ m GL}_3(Q_p)$.

Contribution

It provides a more precise formulation of Breuil's conjecture and proves new cases, advancing understanding of local-global compatibility in p-adic representation theory.

Findings

Proved special cases of the refined conjecture for ${ m GL}_3(Q_p)$.

Reinterpreted Breuil's conjecture using $(, )$-modules over the Robba ring.

Enhanced the understanding of local-global compatibility in the context of p-adic groups.

Abstract

We reinterpret a conjecture of Breuil on the locally analytic $\mathrm{Ext}^1$ in a functorial way using $(\varphi,\Gamma)$-modules (possibly with $t$-torsion) over the Robba ring, making it more accurate. Then we prove several special or partial cases of this "improved" conjecture, notably for ${\rm GL}_3(\mathbb{Q}_p)$.